Foundations of Computer Science Lecture 28 Efficiency: The Class P, NPand NP-Completeness

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Foundations of Computer Science Lecture 28

Efficiency: The Class P, NPand NP-Completeness

Running Time

Efficiently Solvable Problems

Boundary Between Efficient and Inefficient

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Last Time

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Computer: Universal Turing Machine ^U

^tm

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Program and Input: hM i # w . ^U

^tm

simulates ^M on ^w .

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No Program Verifier, no Ultimate-Debugger, no PCP-Solver.

4

No means No.

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The Path Forward: Focus on Decidable Problems

FOCS

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The Path Forward: Focus on Decidable Problems

FOCS

T h e o r y o f C o m p u t in g c r e t e M a t h

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The Path Forward: Focus on Decidable Problems

FOCS

T h e o r y o f C o m p u t in g D is c r e t e M a t h

Decider U_tm=computer TM = Algorithm

CFG Parsing

DFA RegExp

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The Path Forward: Focus on Decidable Problems

FOCS

T h e o r y o f C o m p u t in g c r e t e M a t h

CFG Parsing

DFA RegExp

Proof, logic INDUCTION

Recursion Struct. Induction

Sums, Asymptotics

Number theory

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The Path Forward: Focus on Decidable Problems

FOCS

T h e o r y o f C o m p u t in g D is c r e t e M a t h

CFG Parsing

DFA RegExp

Sums, Asymptotics

Number theory

Graphs

Counting

Probability Multivariate Calc.

Linear Algebra Probability Theory Graph theory

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The Path Forward: Focus on Decidable Problems

FOCS

T h e o r y o f C o m p u t in g c r e t e M a t h

CFG Parsing

DFA RegExp

Sums, Asymptotics

Number theory Multivariate Calc.

FAST (P) Polynomial FAST (NP) Unbounded Parallelism SLOW Exponential Boolean Circuits efficiency

P = NP ?

Chapters 28 & 29

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The Path Forward: Focus on Decidable Problems

FOCS

T h e o r y o f C o m p u t in g D is c r e t e M a t h

CFG Parsing

DFA RegExp

Sums, Asymptotics

Number theory

Graphs

Counting

P = NP ?

Chapters 28 & 29

Introduction to Algorithms Computability&Complexity

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The Path Forward: Focus on Decidable Problems

FOCS

T h e o r y o f C o m p u t in g c r e t e M a t h

CFG Parsing

DFA RegExp

Sums, Asymptotics

Number theory Multivariate Calc.

P = NP ?

Chapters 28 & 29

Introduction to Algorithms Computability&Complexity

Algorithms & DS – Approximation – Randomized – Distributed

Cryptography Data

– ML/AI/DM/NLP – Vision

– Graphics – Comp. Finance

Networks – Computers – Social

– Data (e.g. www) Robotics

Security

Programming Languages – Compilers

– Distributed Program Analysis – Testing

– Verification

Theory Algorithms

AI

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The Path Forward: Focus on Decidable Problems

FOCS

T h e o r y o f C o m p u t in g D is c r e t e M a t h

CFG Parsing

DFA RegExp

Sums, Asymptotics

Number theory

Graphs

Counting

P = NP ?

Chapters 28 & 29

Introduction to Algorithms

Computer Organization

Principles of Software Computability&Complexity

Algorithms & DS – Approximation – Randomized – Distributed

Cryptography Data

– ML/AI/DM/NLP – Vision

– Graphics – Comp. Finance

Networks – Computers – Social

– Data (e.g. www) Robotics

Security

Programming Languages – Compilers

– Distributed Program Analysis – Testing

– Verification DB Systems Parallel computing Operating systems

Architecture

Theory Algorithms

AI

Software Systems

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Today: Efficiency: The Class P, NPand NP-Completeness

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Time complexity: Asymptotic worst-case analysis.

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The class P: Efficiently solvable problems.

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Polynomial on one architecture means polynomial on pretty much any architecture.

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Running Time

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Time Complexity

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Two Tapes vs. One Tape

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Decidable But Non -Polynomial

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Boundary Between Efficient and Inefficient

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. . . the high technology so celebrated today is essentially a mathematical technology.

“To err is human, but to really foul things up you need a computer.” – Paul Ehrlich

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. . . the high technology so celebrated today is essentially a mathematical technology.

“To err is human, but to really foul things up you need a computer.” – Paul Ehrlich

Mariner rocket explodes (1962). Formula into code bug resulted in no smoothing of deviations.

WWWIII (1983)? Soviet EWS detects 5 US-missiles (bug detected sunlight reflections).

◮

Luckily Stanislav “funny feeling in my gut” Petrov thought: “surely they’d use more missiles?”

Therac 25 (1985). Concurrent programming bug killed patients through massive 100× radiation overdose.

AT&T Lines Go Dead (1990). 75 million calls dropped (one line of buggy code in software upgrade).

Patriot missile defense fails (1991). 28 soldiers dead, 100 injured (rounding error in scud-detection).

Pentium floating point long-division bug (1993). Cost: $475 million – flawed division table.

Ariane rocket explosion (1996). Cost: $500 million – overflow in 64-bit to 16-bit conversion.

Y2K (1999). Cost: $500 billion spent because year was stored as 2 digits to save space.

Mars Climate Orbiter Crash (1998). Cost: $125 million lost due to metric to imperial units bug.

Tesla Self-Driving Car (2016). 1 dead. Auto-pilot didn’t “see” tractor-trailer.

Financial Disasters: London Stock Exchange down due to single server bug (2009; billions of pounds of trading);

Knight Capital computer glitch trigers stock sale (2012; 500 million lost and Knight’s value drops by 75%).

Airline Disasters:

^◮ AirFrance 447 2009, 228 dead: pitot-tube failure feeds inconsistent data to programs which then panic pilot.

◮ Spanair 5022, 2008, 154 dead: malware virus.

◮ AdamAir 574, 2007, 102 dead: navigation system errors (and pilot errors).

◮ KoreanAir 801, 1997, 228 dead: ground proximity warning system bug.

◮ AeroPerú 603, 1996, 70 dead: altimeter failures.

◮ Scottish RAF Chinook, 1994, 29 dead: faulty test program

◮ AirFrance 296, 1988, 3 dead: altimeter bug.

◮ IranAir 655, 1988, 290 dead: shot down by US Aegis combat system (misidentified as attacking military plane).

◮ KoreanAir 007, 1983, 269 dead: autopilot took plane into Soviet airspace where it got shot down.

◮ Boeing 737 Max, 2018,2019, 346 dead: attack sensor + algorithm errors.

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. . . the high technology so celebrated today is essentially a mathematical technology.

“To err is human, but to really foul things up you need a computer.” – Paul Ehrlich

Mariner rocket explodes (1962). Formula into code bug resulted in no smoothing of deviations.

WWWIII (1983)? Soviet EWS detects 5 US-missiles (bug detected sunlight reflections).

◮

Luckily Stanislav “funny feeling in my gut” Petrov thought: “surely they’d use more missiles?”

Therac 25 (1985). Concurrent programming bug killed patients through massive 100× radiation overdose.

AT&T Lines Go Dead (1990). 75 million calls dropped (one line of buggy code in software upgrade).

Patriot missile defense fails (1991). 28 soldiers dead, 100 injured (rounding error in scud-detection).

Pentium floating point long-division bug (1993). Cost: $475 million – flawed division table.

Ariane rocket explosion (1996). Cost: $500 million – overflow in 64-bit to 16-bit conversion.

Y2K (1999). Cost: $500 billion spent because year was stored as 2 digits to save space.

Mars Climate Orbiter Crash (1998). Cost: $125 million lost due to metric to imperial units bug.

Tesla Self-Driving Car (2016). 1 dead. Auto-pilot didn’t “see” tractor-trailer.

Financial Disasters: London Stock Exchange down due to single server bug (2009; billions of pounds of trading);

Knight Capital computer glitch trigers stock sale (2012; 500 million lost and Knight’s value drops by 75%).

Airline Disasters:

^◮ AirFrance 447 2009, 228 dead: pitot-tube failure feeds inconsistent data to programs which then panic pilot.

◮ Spanair 5022, 2008, 154 dead: malware virus.

◮ AdamAir 574, 2007, 102 dead: navigation system errors (and pilot errors).

◮ KoreanAir 801, 1997, 228 dead: ground proximity warning system bug.

◮ AeroPerú 603, 1996, 70 dead: altimeter failures.

◮ Scottish RAF Chinook, 1994, 29 dead: faulty test program

Foundations of Computer Science Lecture 28 Efficiency: The Class P, NPand NP-Completeness